Microreflection delay estimation in a CATV network

CROSS-REFERENCE TO RELATED APPLICATIONS

0001None

BACKGROUND

0002The present disclosure relates to systems and methods that provide optical signals over a cable transmission network.

0003Cable television (CATV) networks have evolved significantly since first being deployed as relatively simple systems that delivered video channels one-way from a content provider. These early systems included transmitters that split a number of CATV channels among several frequency bands, each of approximately 6 MHz, time multiplexed those signals, and sent them to subscribers as an electrical signal through a network of coaxial transmission lines to cable modems or set-top boxes in subscribers homes. Early evolution of such systems permitted limited return communication from the subscribers back to the content provider either through telephone lines or a dedicated, small, low-frequency signal propagated onto the coaxial network.

0004Modern CATV networks differ from these early networks in several material respects. First, given the growth of content provided to customers over increasing distances, much of the coaxial transmission path has been replaced by fiber-optic lines, although the trunk lines and feeder cables to customer homes remain coaxial, and much of the content delivered to the provider for transmission is by an analog electrical signal that the provider must then convert to an optical signal for propagation onto the coaxial transmission grid. Nodes, located closer to the subscribers receive the optical signals from the transmitter and convert them to the electrical signals for delivery to the subscribers through the coaxial lines to the subscribers' homes. Such a network is commonly referred to as a Hybrid Fiber-Coax (HFC) system.

0005Second, in addition to delivering television content, modern HFC systems also deliver additional content including Video-on Demand (VOD) and data services such as Internet access. These services require not only a high-capacity transmission path in the direction from the content provider to the subscriber but require a high capacity transmission path in the direction from the subscriber back to the content provider. Additional quality is needed both for new high definition broadcast channels and for the narrowcast video and data services. The original HFC network has been successfully updated to deliver new services, but the pressure of HD, VOD, and data service requires further advances, thus there is significant continuing pressure to increase the quality of data sent over a HFC network.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

0006<figref idref="DRAWINGS">FIG. 1</figref> shows an exemplary source of micro-reflection impairment.

0007<figref idref="DRAWINGS">FIG. 2</figref> shows an ideal constellation for a 16-QAM signal.

0008<figref idref="DRAWINGS">FIG. 3</figref> shows a constellation for a 16-QAM signal that is impaired by micro-reflection.

0009<figref idref="DRAWINGS">FIG. 4</figref> shows a system capable of mitigating micro-reflections.

0010<figref idref="DRAWINGS">FIG. 5</figref> shows an exemplary impulse response associated with negligible micro-reflection impairment

0011<figref idref="DRAWINGS">FIG. 6</figref> shows a Fast Fourier Transform (FFT) of the response of <figref idref="DRAWINGS">FIG. 5</figref>.

0012<figref idref="DRAWINGS">FIG. 7</figref> shows an exemplary impulse response associated with micro-reflection impairment.

0013<figref idref="DRAWINGS">FIG. 8</figref> shows the pre-equalization channel frequency response to micro-reflection impairment.

0014<figref idref="DRAWINGS">FIG. 9</figref> shows a first estimated sinusoidal micro-reflection waveform.

0015<figref idref="DRAWINGS">FIG. 10</figref> shows a second estimated sinusoidal micro-reflection waveform.

0016<figref idref="DRAWINGS">FIGS. 11-13</figref> each show simulation results for variance of a respective variable in an estimated sinusoidal micro-reflection impairment as a function of data records.

0017<figref idref="DRAWINGS">FIGS. 14-17</figref> each show simulation results for variance of a respective variable in an estimated sinusoidal micro-reflection impairment as a function of SNR.

0018<figref idref="DRAWINGS">FIG. 18</figref> shows an example of a pre-equalizer capable of modifying a signal to compensate for micro-reflections.

0019<figref idref="DRAWINGS">FIG. 19</figref> shows a response of the pre-equalizer of <figref idref="DRAWINGS">FIG. 18</figref> to an impulse signal.

0020<figref idref="DRAWINGS">FIG. 20</figref> shows a pole-zero plot of an exemplary response of a pre-equalizer.

0021<figref idref="DRAWINGS">FIG. 21</figref> shows a pole-zero plot of a modified response of <figref idref="DRAWINGS">FIG. 20</figref>.

0022<figref idref="DRAWINGS">FIG. 22</figref> shows the frequency response of an inverted impulse response of a pre-equalizer.

0023<figref idref="DRAWINGS">FIG. 23</figref> shows an interpolation technique to improve an estimate of micro-reflection delay.

0024<figref idref="DRAWINGS">FIG. 24</figref> shows a time-domain method of calculating the distance to a micro-reflection impairment based on the coefficients of a pre-equalizer.

DETAILED DESCRIPTION

0025Physically, existing downstream CATV content originates on the network from a head end, within which is a Cable Modem Termination System (CMTS). In the CMTS is typically a plurality of EdgeQAM units, which each receive encoded video signals, audio signals, and/or IP signals, and which each directly output a spectrum of amplitude-modulated analog signals at a defined frequency or set of frequencies to an RF combining network. The RF combining network in turn delivers the combined, modulated signals to a transmitter that converts the analog signals to optical signals and propagates those signals to a fiber optic network along a forward path to a receiver in a node. The node, in turn, converts the received optical signals back into electrical signals so that they may be delivered to subscribers through coaxial cables that terminate in the subscriber's homes. The node, conversely will also have equipment capable of receiving electrical signals from the subscribers, converting them to optical signals, and sending the optical signals to the CMTS along the fiber optic network, where the CMTS has a receiver capable of decoding and converting the signals back to the analog domain, respectively.

0026In the specification, the drawings, and the claims, the terms “forward path” and “downstream” may be interchangeably used to refer to a path from a head end to a node, a node to an end-user, or a head end to an end user. Conversely, the terms “return path”, “reverse path” and “upstream” may be interchangeably used to refer to a path from an end user to a node, a node to a head end, or an end user to a head end. Also, it should be understood that, unless stated otherwise, the term “head end” will also encompass a “hub,” which is a smaller signal generation unit downstream from a head end, often used for community access channel insertion and other purposes, that generally mimics the functionality of a head end, but may typically not include equipment such as satellite dishes and telephone equipment.

0027Directly-modulated EdgeQAM units have become increasingly sophisticated, offering successively higher densities, which in turn means that each EdgeQAM unit can process more channels of CATV data. For example, modern EdgeQAM modulation products can now simultaneously generate 32 or more channels on a single output port. With more channels being modulated per output port, the amount of combining required by the RF combining network <b>350</b> is reduced, with a corresponding simplification in the circuitry at the head end. The term ‘QAM’ is often used to interchangeably represent either: (1) a single channel typically 6 MHz wide that is Quadrature Amplitude Modulated (thus a “32 QAM system” is shorthand for a system with 32 Quadrature Amplitude Modulated channels; or (2) the depth of modulation used by the Quadrature Amplitude Modulation on a particular channel, e.g. 256 QAM means the signal is modulated to carry 8 bits per symbol while 4096 QAM means the signal is modulated to carry 12 bits per symbol. A higher QAM channel count or a higher QAM modulation means that a higher number of content “channels” can be delivered over a transmission network at a given standard of quality for audio, video, data, etc. QAM channels are constructed to be 6 MHz in bandwidth in North America, to be compatible with legacy analog TV channels and other existing CATV signals. However, more than one video program or cable modem system data stream may be digitally encoded within a single QAM channel. The term channel is unfortunately often used interchangeably, even though a QAM channel and a video program are not often the same entity—multiple video programs can be and usually are encoded within a single 6 MHz QAM channel. In this case, the modern EdgeQAM modulation products generate multiple instances of the 6 MHz bandwidth QAM channels. This simplifies the head end structure since some subset of the RF combining is now performed within the EdgeQAM units rather than in the external RF combining network. Packaging multiple QAM generators within a single package also offers some economic value.

0028One existing source of impairment on the quality of signals sent between components in an HFC network is micro-reflections. Micro-reflections are generally defined as digital signal copies, which are delayed in time due to traveling a path longer than that intended. Micro-reflections are analogous to echoes within the CATV network, and though they can manifest within CATV networks in a variety of ways, they are mostly due to faulty equipment that also reflect a signal, rather than simply propagating it in a desired direction. <figref idref="DRAWINGS">FIG. 1</figref> illustrates one example of how a micro-reflection can occur in a network. This figure shows a transmission path between a CMTS <b>10</b> in a head end and a cable modem <b>12</b> in a subscriber's home. In this figure, a signal <b>16</b> is sent along a return path from the cable modem <b>12</b> to the head end <b>10</b>, but is reflected by a defective component <b>14</b>, typically a damaged cable, or a corroded or loose connector, essentially exhibiting the properties of a less than ideal match. A portion of the digital signal then propagates back along path <b>18</b> toward the cable modem <b>12</b>, which then reflects the signal again. The cable modem <b>12</b> is designed to support at least a 6 dB return loss. The 6 dB return loss equates to approximately 25% of the incident signal power that will be reflected. In the CATV network, design tolerances typically produce much better results, but over time performance will degrade as the plant gets older and elements that contribute to good RF matching degrade. The total delay associated with a micro-reflection is directly related to the length of cable(s) associated with the micro-reflection path.

0029Micro-Reflections are commonly considered within the CATV industry as linear distortion, meaning that the impairment will change by one decibel for a decibel change in the digital signal level. The effect that micro-reflections have on digital signals is Inter-Symbol Interference (ISI). When viewing a demodulated constellation impaired by a micro-reflection, the digital symbols will be spread from their ideal constellation points, as illustrated by <figref idref="DRAWINGS">FIGS. 2 and 3</figref>. <figref idref="DRAWINGS">FIG. 2</figref> illustrates an ideal constellation for a 16-QAM signal. The pattern in which the individual symbols will spread can be observed as a miniaturized constellation in very low noise systems. <figref idref="DRAWINGS">FIG. 3</figref>, however, illustrates a 16-QAM constellation impaired by Micro-Reflection whose relative amplitude is −20 dBc and delay is 4 symbol periods.

0030<figref idref="DRAWINGS">FIG. 4</figref> shows an illustrative system <b>20</b> capable of detecting and mitigating micro-detected reflections. Without mitigation, the micro-reflection will cause in-band ripples on the signal's frequency response. The system <b>20</b> may include a head end <b>22</b> connected to one or more cable modems <b>25</b> through a node <b>24</b> via a network comprising some combination of fiber-optic cables <b>21</b> and coaxial cables <b>23</b>. The head end <b>22</b> may include a plurality of EdgeQAM units <b>26</b> capable of modulating respective content signals (i.e. video, audio, and data content) to be delivered to customers over a forward path from the head end <b>22</b> to the modems <b>25</b>. The EdgeQAM units <b>26</b> output the respective content signals to a RF combining network <b>28</b> which may multiplex the signals and propagate the content to the respective cable modems by a transmitter <b>30</b>, such as a laser transmitter, which transmits a signal to a receiver <b>31</b> in the node <b>24</b>. The node <b>24</b> then forwards the demodulated signals to the respective cable modems <b>25</b> through diplexers <b>34</b>. The system <b>20</b> is also preferably capable of delivering return path signals from the cable modems <b>25</b> to the head end <b>22</b> through the node <b>24</b> using the node's transmitter <b>32</b> and the head end's receiver <b>33</b>.

0031As noted previously, the HFC network interconnecting the head end <b>22</b> with the modems <b>25</b> may include one or more sources of micro-reflections that distort the signals passing through the network. Thus, the system <b>20</b> preferably corrects for those micro-reflections in that the head end <b>22</b> may receive and measure a signal from one or more of the modems <b>25</b> so as to determine coefficients with which to bias, or pre-distort, signals transmitted over the network so that the pre-distortion in the signal cancels the distortion of the network caused by the micro-reflections. For example, one or more modems <b>25</b> may transmit an impulse signal to the receiver <b>33</b> of the head end <b>22</b>. The head end <b>22</b> may include a processor <b>38</b> that measures the amount of distortion between the received signal and an impulse signal to calculate coefficients that a pre-equalizer can use to pre-distort the signal. The coefficients may be transmitted to one or more modems <b>25</b>, which each have a respective pre-equalizer <b>39</b>. The respective pre-equalizers <b>39</b> of the modems <b>25</b> may then pre-distort any subsequent signals sent through the HFC network, based on the coefficients received from the head end <b>22</b>, such that they arrive at the head end <b>22</b> less distorted than they would be without pre-equalization. Alternatively (or additionally), the head end <b>22</b> may include its own pre-equalizer <b>36</b> with which signals passing between the head end <b>22</b> and the modems <b>25</b> may be pre-distorted to compensate for microreflections. Techniques for pre-equalization have been disclosed in U.S. Pat. No. 8,537,972, issued on Sep. 17, 2013 to Thompson et al., the disclosure of which is hereby incorporated by reference.

0032<figref idref="DRAWINGS">FIGS. 5-8</figref> show pre-equalization responses where there is, and is not respectively, micro-reflection in the signal received by a CMTS. Specifically, <figref idref="DRAWINGS">FIG. 5</figref> shows a cable modem transmission impulse response magnitude associated with negligible impairment. In this example, the equalizer's main tap <b>40</b> is ideally 0 dB, while the non main taps are appreciably lower. Non main tap values will vary with system noise in the absence of other impairments. For the case in <figref idref="DRAWINGS">FIG. 5</figref>, the system noise is likely between 40-50 dB resulting in the non-main tap values to exhibit what appears to be random values falling within this range. <figref idref="DRAWINGS">FIG. 6</figref>, which shows a Fast Fourier Transform (FFT) of the equalizer coefficients of <figref idref="DRAWINGS">FIG. 5</figref>, reveals that the frequency response magnitude is relatively constant when there is negligible impairment. Thus, when a CMTS receives a signal similar to that of <figref idref="DRAWINGS">FIGS. 5 and 6</figref>, there is not likely to be any micro-reflection impairment.

0033<figref idref="DRAWINGS">FIGS. 7 and 8</figref> both illustrate the effect the previously discussed micro-reflection has on both the time and frequency domain magnitude responses. The micro-reflection characteristics, relative amplitude=−20 dBc and delay=4*T can be readily observed from the impulse response of <figref idref="DRAWINGS">FIG. 7</figref>. Because micro-reflections always have a delay characteristic, energy used to cancel micro-reflections will always be within the post-tap region of the pre-equalizer (taps <b>0</b> through <b>16</b>).

0034Though the system <b>20</b> is capable of detecting micro-reflection induced delays in a signal, and mitigating them through a pre-equalization bias, the system <b>20</b> does not determine the location of the micro-reflection which is necessary for e.g. repair. Accurately estimating the location of micro-reflections is a common problem in the CATV industry. Though in general terms, micro-reflection delay can be roughly translated into cable distance by the velocity of signal propagation through the transmission system, distances calculated from the delay are not sufficiently accurate to pinpoint the precise location of the impairment, necessitating costly and time-consuming transport and connection of test equipment over a broad area of the HFC network.

0035The present disclosure presents more effective techniques for determining the location of a micro-reflection impairment from a pre-equalization response. At the outset, equations (1) through (5) generally define the relationship between amplitude ripple and the level of a micro-reflection.

0036<maths id="MATH-US-00001" num="00001"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>Γ</mi><mi>E</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>Z</mi><mi>E</mi></msub><mo>-</mo><msub><mi>Z</mi><mn>0</mn></msub></mrow><mrow><msub><mi>Z</mi><mi>E</mi></msub><mo>+</mo><msub><mi>Z</mi><mn>0</mn></msub></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>VSWR</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><msub><mi>Γ</mi><mi>E</mi></msub></mrow><mrow><mn>1</mn><mo>-</mo><msub><mi>Γ</mi><mi>E</mi></msub></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mi>Ripple</mi><mo>⁡</mo><mrow><mo>(</mo><mi>dB</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mn>10</mn><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mrow><msub><mi>log</mi><mn>10</mn></msub><mo>⁡</mo><mrow><mo>(</mo><msup><mi>VSWR</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Γ</mi><mi>E</mi></msub><mo>=</mo><mfrac><mrow><mi>VSWR</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>VSWR</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mi>MicroreflectionLevel</mi><mo>⁡</mo><mrow><mo>(</mo><mi>dB</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mn>10</mn><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mrow><msub><mi>log</mi><mn>10</mn></msub><mo>⁡</mo><mrow><mo>(</mo><msubsup><mi>Γ</mi><mi>E</mi><mn>2</mn></msubsup><mo>)</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> Using the equations provided in [5], one can readily verify that −20 dBc of micro-reflection level equates to approximately 0.87 dB of micro-reflection ripple.

0037It can also be seen from <figref idref="DRAWINGS">FIG. 8</figref>, that there are four complete waveforms within the channel frequency response resulting from a total micro-reflection delay of four symbol periods or 4*T. Given that the velocity of propagation, VoP, is commonly specified by cable manufacturers, estimates for cable length may be made. For example, modern foam dielectric trunk and feeder cables have a VoP=87%. This equates to 1.17 ns/ft of transit delay. Therefore, the length of cable associated with symbol rate=5.12 Msps, and a micro-reflection delay=4*T will result in a cable length of approximately 334 feet of cable. Because micro-reflection delay is actually round trip delay, the cable length should be based on half of the total micro-reflection delay.

0038Problems with estimating micro-reflection delay arise when micro-reflection delays are not exactly symbol spaced. For a symbol rate of 5.12 Msps and VoP of 87%, each Pre-Equalizer tap represents approximately 83.5 feet. If one were to rely solely on selecting the Pre-Equalizer tap with the highest energy to represent the micro-reflection delay, cable length estimation error could be as high as ±83.5 feet. This amount of estimation error is unacceptable and could result in mistakenly identifying many suspect cables during the micro-reflection isolation process. A more precise estimate of the cable length associated with a micro-reflection would better enable cable repair technicians to identify suspect lengths of cable contributing to micro-reflection impairment.

0039One solution proposed by the present disclosure begins by first fitting a sinusoidal waveform against the observed micro-reflection waveform using Cramer-Rao Lower Bound (CRLB) for Sinusoidal Parameter Estimation technique: <br /><i>x[n]=A </i>cos(2π<i>f</i>₀<i>n</i>+φ)+<i>w[n]</i><br /><i>n=</i>0,1, . . . ,<i>N−</i>1  (6)<br /> Estimation of all three sinusoidal parameters; frequency, amplitude and phase will be required.

0040The variation associated with amplitude, frequency, and phase relative to the CRLB will provide valuable insight to estimation performance. Additionally, variance assessment will also aid in guiding appropriate values for N and SNR, for which micro-reflection estimations may be made efficiently and accurately. The value for examining estimation variance as a function of N is significant since increasing N could appreciably add to required estimation processing and ultimately cost.

0041For an initial evaluation, the waveforms were tabulated using Microsoft Excel. N=64, and 1024 are illustrated in <figref idref="DRAWINGS">FIGS. 10, and 11</figref> respectively where <figref idref="DRAWINGS">FIG. 10</figref> assumed that N in equation (6) is 64 while <figref idref="DRAWINGS">FIG. 11</figref> assumed that N was 1024. Variance, σ², was calculated using Equation 7, with SNR=10 dB.

0042<maths id="MATH-US-00002" num="00002"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>SNR</mi><mo>=</mo><mrow><mi>η</mi><mo>=</mo><mfrac><msup><mi>A</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>⁢</mo><msup><mi>σ</mi><mn>2</mn></msup></mrow></mfrac></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> The signal parameters used were as previously described, A=0.87 dB, f₀=4 Hz, and φ=π/2 radians. The Microsoft Excel add-in “Solver” was used to minimize error between the observed signal s[n] <b>42</b> embedded in noise x[n] <b>44</b>, and a signal guess ŝ[n] <b>46</b> that closely followed the signal s[n] <b>42</b>, varying amplitude, frequency and phase simultaneously. Microsoft Excel add-in “solver” would not estimate the true parameter value reliably unless the initial conditions were already at values close to the true parameters.

0043The waveform <b>41</b> in <figref idref="DRAWINGS">FIGS. 9 and 10</figref> represents the square error, based upon the Least Squares Error Criterion from Equation 8.

0044<maths id="MATH-US-00003" num="00003"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mi>J</mi><mo>⁡</mo><mrow><mo>(</mo><mi>ϑ</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><msup><mrow><mo>(</mo><mrow><mrow><mi>x</mi><mo>⁡</mo><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mo>-</mo><mrow><mi>s</mi><mo>⁡</mo><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> Comparing <figref idref="DRAWINGS">FIGS. 9 and 10</figref>, there is appreciable difference between the square error. N=64 square error had values that ranged 4-6, while N=1024 square error ranged between 64-96. This makes sense since the N=1024 signal has 16 times as many points as the N=64 signal and each point for both cases has the same variation due to noise.

0045A Matlab model was also used in a similar fashion to what has been illustrated using Microsoft Excel. The waveform s[n] was generated using Equation 6 and the true parameters originally specified. The waveform w[n] was generated via specified SNR, amplitude, A, and variance, σ², using Equation 7. A key difference between the two models, however, was the way in which the guesses were generated. Microsoft Excel offered no control over how amplitude, frequency, and phase were varied. However, within Matlab, the three parameters were part of a nested “for” loop series sweeping over the following specified ranges. <ul id="ul0001" list-style="none"><li id="ul0001-0001" num="0000"><ul id="ul0002" list-style="none"><li id="ul0002-0001" num="0046">A=0:0.1:5.0</li><li id="ul0002-0002" num="0047">f₀=0:0.01:16</li><li id="ul0002-0003" num="0048">φ=0:π/4:2 π <br /> Each loop creates an estimate, g[n], using the above parameters to compare to the observed signal x[n] for Least Squares, LS, error as described in Equation 8. Amplitude ripple scenarios were limited between 0 and 5 dB. As previously discussed 5 dB of micro-reflection ripple equates to approximately −11 dBc relative to the main tap energy. This magnitude micro-reflection is not a common occurrence in today's CATV networks, but certainly a condition in which the equalizer could correct for. A more complete range for A could be 0 to 15 dB of ripple or −3 dBc level, beyond which any higher energy in the equalizer tap would likely result in equalizer instability and failure. </li></ul></li></ul>

0049The frequency range was limited to 16 Hz. Because the frequency within the digital signal channel response is directly related to the micro-reflection delay path, frequency (and subsequently delay) resolution of 1/100^thof a Hz was chosen to achieve estimation of suspect cable lengths to within ±0.835 ft accuracy. It was previously noted that whole frequency, or delay accuracy equated to approximately ±83.5 ft accuracy in cable length, assuming R_s=5.12 Msps and VoP=87%. Further, 1/10^thtap accuracy improves cable length estimation accuracy to ±8.35 ft; and finally, ±0.835 ft accuracy with 1/100^thtap. Of course, as the R_sand VoP value change, so will the accuracy of the micro-reflection cable length estimator. For example, reducing to R_s=2.56 Msps, degrades the accuracy of the model to ±1.67 ft, assuming all other aspects of the estimator remain equal. Finally, the range and resolution selection for phase preferably ensures optimal estimation of frequency.

0050Once all of the loops have been executed, the set of parameters that result in minimal LS error, J(Θ), is chosen as the estimated signal, ŝ[n], representing estimates for amplitude, frequency, and phase.

0051CRLB assessment as a function of N was facilitated using Monte Carlo simulations in Matlab, where the number of Monte Carlo runs M per value of N, was 100. 10 different N values were evaluated each being a power 2, i.e. N=[2, 4, 8, . . . , 1024]. CRLB for each parameter is represented in equations 9 through 11, for amplitude, frequency, and phase.

0052<maths id="MATH-US-00004" num="00004"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mi>var</mi><mo>⁡</mo><mrow><mo>(</mo><mover><mi>A</mi><mo>^</mo></mover><mo>)</mo></mrow></mrow><mo>≥</mo><mrow><mo>(</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><msup><mi>σ</mi><mn>2</mn></msup></mrow><mi>N</mi></mfrac><mo>)</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>9</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mi>var</mi><mo>⁡</mo><mrow><mo>(</mo><msub><mover><mi>f</mi><mo>^</mo></mover><mn>0</mn></msub><mo>)</mo></mrow></mrow><mo>≥</mo><mrow><mo>(</mo><mfrac><mn>12</mn><mrow><msup><mrow><mo>(</mo><mrow><mn>2</mn><mo>⁢</mo><mi>π</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>⁢</mo><mi>η</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mrow><mi>N</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><msup><mi>N</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo>)</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>10</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mi>var</mi><mo>⁡</mo><mrow><mo>(</mo><mover><mi>ϕ</mi><mo>^</mo></mover><mo>)</mo></mrow></mrow><mo>≥</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo>⁢</mo><mi>N</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi>η</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mrow><mi>N</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>N</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>11</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

0053Simulation results are presented in <figref idref="DRAWINGS">FIGS. 11 through 13</figref>, where the curve <b>50</b> represents the CRLB from equations 9 through 11 and the curve <b>52</b> represents variance of M=100 Monte Carlo simulations. These figures show how the estimator variance improves as N increases. For N>32, estimator variance improves substantially. Many simulations will be used to evaluate estimation performance against SNR, therefore N=64 will be selected. With N=64, some estimator performance will be lost through increased variation, but is something worth tolerating in order to achieve results within a reasonable time frame.

0054Simulation distribution for each parameter estimate as a function of SNR is illustrated in <figref idref="DRAWINGS">FIGS. 14 through 16</figref>, where the SNR values are specified in the legend. The effect of improving SNR can readily be seen for the amplitude and frequency estimates.

0055The phase distributions do not have the same resolution as amplitude and frequency, and may be the likely cause of why the observed effect of SNR does not gradually change from one SNR value to the other like it does for amplitude and frequency. Additionally, there appears to be a value of SNR in which the phase estimation exhibits a cliff effect, where appreciably worse estimation occurs over a shorter range of SNR. This SNR value is likely between 0 and 10 dB.

0056<figref idref="DRAWINGS">FIG. 17</figref> shows that overall, the variance associated with estimating amplitude, frequency, and phase improves appreciably with SNR>10 dB. This is beneficial since most CATV upstream systems are designed and maintained to support 16-QAM signaling, which requires SNR≧22 dB for gray encoded BER=1E-8 [7] a typical operating range for most cable system operators.

0057Table 1 includes results where each SNR entry is based upon M=100 Monte Carlo simulations. Estimation of frequency, the key parameter for estimating cable lengths associated with micro-reflections, converges with SNR as low as 5 dB. Amplitude exhibits 0.03 dB of error because the resolution of the estimator is 0.1 dB. This error could be eliminated with increased resolution at the expense of estimation processing requirements. However, 0.01 dB of amplitude ripple equates to a relative micro-reflection level of −64.80 dBc, which is of little consequence since differences this small would likely be buried well below the effects of system noise.

0058<tables id="TABLE-US-00001" num="00001"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="1"><colspec colname="1" colwidth="217pt" align="center" /><thead><row><entry namest="1" nameend="1" rowsep="1">TABLE 1</entry></row></thead><tbody valign="top"><row><entry namest="1" nameend="1" align="center" rowsep="1" /></row><row><entry>ESTIMATED MEAN FOR M =</entry></row><row><entry>100 MONTE CARLO SIMULATIONS</entry></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="5"><colspec colname="offset" colwidth="21pt" align="left" /><colspec colname="1" colwidth="21pt" align="center" /><colspec colname="2" colwidth="77pt" align="center" /><colspec colname="3" colwidth="35pt" align="center" /><colspec colname="4" colwidth="63pt" align="center" /><tbody valign="top"><row><entry /><entry>SNR</entry><entry>Amplitude</entry><entry>Frequency</entry><entry>Phase</entry></row><row><entry /><entry namest="offset" nameend="4" align="center" rowsep="1" /></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="5"><colspec colname="offset" colwidth="21pt" align="left" /><colspec colname="1" colwidth="21pt" align="char" char="." /><colspec colname="2" colwidth="77pt" align="char" char="." /><colspec colname="3" colwidth="35pt" align="char" char="." /><colspec colname="4" colwidth="63pt" align="char" char="." /><tbody valign="top"><row><entry /><entry>0.00</entry><entry>0.90</entry><entry>4.02</entry><entry>1.47</entry></row><row><entry /><entry>2.50</entry><entry>0.88</entry><entry>4.01</entry><entry>1.55</entry></row><row><entry /><entry>5.00</entry><entry>0.88</entry><entry>4.00</entry><entry>1.59</entry></row><row><entry /><entry>7.50</entry><entry>0.87</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>10.00</entry><entry>0.87</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>12.50</entry><entry>0.88</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>15.00</entry><entry>0.87</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>17.50</entry><entry>0.89</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>20.00</entry><entry>0.89</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>22.50</entry><entry>0.89</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>25.00</entry><entry>0.90</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>27.50</entry><entry>0.90</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>30.00</entry><entry>0.90</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>32.50</entry><entry>0.90</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>35.00</entry><entry>0.90</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>37.50</entry><entry>0.90</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>40.00</entry><entry>0.90</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>42.50</entry><entry>0.90</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>45.00</entry><entry>0.90</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>47.50</entry><entry>0.90</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry>50.00</entry><entry>0.90</entry><entry>4.00</entry><entry>1.57</entry></row><row><entry /><entry namest="offset" nameend="4" align="center" rowsep="1" /></row></tbody></tgroup></table></tables>

0059The foregoing estimation technique is performed in the frequency domain, utilizing a least square error approach for estimating a delay due to a micro-reflection, however the disclosure is not limited to this approach. In some instances, for example, it may be preferable to use a time-based approach that directly analyzes a pre-equalizer's converged set of coefficients, and applies a modification to those coefficients to better estimate the micro-reflection delay.

0060A signal Y[n] received over a HFC network having one or more micro-reflections may be modeled in a discrete-time domain as the sum of the transmitted signal X[n] and one or more attenuated copies, with respective delays, as described in equation 12. <br /><i>Y[n]=X[n]+α</i>₁<i>X[n−</i>1]+α₂<i>X[n−</i>2]+ . . . α_m<i>X[n−m]</i>  (12)<br /> The respective micro-reflections may result from several distinct faults in the transmission path, causing what is called a multiple recursion echo, or may result from a single fault in the transmission network traveling along one of several paths to the destination, which is called a single recursion echo. Regardless of the type of echo, a properly designed pre-equalization system can compensate for either type, by using a multiple-tap pre-equalizer to approximate an inverse response to the distortion produced by the combination of echoes. If the echoes are weak, the first recursion echo is the only significant echo in either the single or multiple recursion case, and the first tap of the pre-equalizer will compensate for most of the distortion. However, even in unique echo cases like the single-transit echo, the multiple taps in the pre-equalizer can be used to adapt to the echo, a transfer function describing a CATV network communication channel A(z) will be comprised of many delay elements.

0061Assuming that X[n] of equation 12 is an impulse function, Equation 13 is z-domain representation A(z) of the signal Y[n]. <br /><i>A</i>(<i>z</i>)=1+α₁<i>z</i>⁻¹+α₂<i>z</i>⁻²+ . . . α_m<i>z</i>^−m  (13)<br /> An Finite Impulse Response (FIR) filter or equalizer implementation will be an approximation of the micro-reflection impairment with a response B(z), whose characteristics ideally match the inverse of the micro-reflection or channel's response A(z) so that when the signal passes through this filter, the effects of the micro-reflection is cancelled and the net response becomes ideally 1 as illustrated in Equation 14 and 15 <br /><i>B</i>(<i>z</i>)≈<i>A</i>(<i>z</i>)⁻¹  (14)<br /><i>Y</i>(<i>z</i>)=<i>A</i>(<i>z</i>)<i>B</i>(<i>z</i>)<i>X</i>(<i>z</i>)  (15)<br /> where A(z)B(z) is equal to 1 and X(z) is an impulse in the Z-domain.

0062In the time domain, the delay associated with a micro-reflection may preferably be based on an pre-equalizer's z-domain impulse response B(z), having an associated time-domain impulse response b[n]. For example, <figref idref="DRAWINGS">FIG. 18</figref> schematically shows a pre-equalizer <b>60</b> having delay elements <b>62</b> after respective taps <b>63</b>, each tap weighted by weights <b>64</b> which correspond to respective pre-equalization coefficients. The respectively delayed taps are summed by elements to produce a time-domain response <b>66</b>. The pre-equalizer of <figref idref="DRAWINGS">FIG. 18</figref> has a number of taps equal to n+m, where n is the number of taps before the main tap b₀, and m is the number of taps after the main tap b₀. It should be understood by those skilled in the art that n and m may be varied as desired, depending on how many delay elements are desired before the main tap, and how many delay elements are desired after the main tap.

0063In some embodiments, a pre-equalizer may be implemented using an AGILENT MXA Series N9020A Vector Signal Analyzer (VSA) configured in DEMODULATION Mode of Operation with the Equalizer Function enabled, and operated using the AGILENT 89600 VSA software. In such a configuration, the AGILENT VSA may implement a pre-equalizer having a desired number of taps (up to 99). For instance, the AGILENT VSA may implement a 39-tap equalizer having a main tap at tap <b>16</b>, and in such a configuration the equalizer may conveniently be truncated to a 24-tap equalizer by simply using the 7 taps prior to the main tap and the 16 taps following the main tap. Alternatively, the pre-equalizer may have any other desired number of taps and/or any other desired split of taps before and after the main tap. Each of the delays and weights may be varied to produce a pre-equalization signal calculated to counteract a detected delay from a micro-reflection. Thus, when a signal from a cable modem or other device encounters micro-reflections within their transmission path to the CMTS, the pre-equalizer <b>60</b> may be used to mitigate the micro-reflection impairment by biasing the device's transmit signal with a micro-reflection of equal amplitude and opposite phase, such that when the CMTS receives the digital signal, the effects of the in-channel micro-reflection impairment have been cancelled by the signal bias.

0064Typically complex coefficients are provided when cable modems are queried for their pre-equalization information. Complex coefficients for a 39-tap equalizer, with a main tap at tap <b>16</b>, have been provided below in order to illustrate the disclosed method of estimating a micro-reflection delay in the time domain.

0065<maths id="MATH-US-00005" num="00005"><math overflow="scroll"><mrow><msub><mi>Coeff</mi><mrow><mi>b</mi><mo>⁡</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msub><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>-</mo><mn>0.00003665078</mn></mrow><mo>-</mo><mrow><mn>0.000003529535</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.0009227422</mn></mrow><mo>-</mo><mrow><mn>0.000011550291</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>-</mo><mn>0.0001179332</mn><mo>-</mo><mrow><mn>0.000007865431</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.0002172839</mn></mrow><mo>-</mo><mrow><mn>0.00004264679</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>-</mo><mn>0.0002135056</mn><mo>-</mo><mrow><mn>0.00002098886</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.0008742874</mn></mrow><mo>-</mo><mrow><mn>0.00007301251</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.0004928805</mn></mrow><mo>+</mo><mrow><mn>0.00003648677</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>-</mo><mn>0.0006369746</mn><mo>-</mo><mrow><mn>0.0003050625</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.0008179067</mn></mrow><mo>+</mo><mrow><mn>0.00009856719</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>-</mo><mn>0.0005863899</mn><mo>-</mo><mrow><mn>0.00004911557</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>-</mo><mn>0.0002600013</mn><mo>-</mo><mrow><mn>0.0001598964</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.002785664</mn></mrow><mo>+</mo><mrow><mn>0.000340097</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>-</mo><mn>0.0055957</mn><mo>-</mo><mrow><mn>0.0009939339</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.0099139</mn></mrow><mo>+</mo><mrow><mn>0.001181552</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>-</mo><mn>0.02338727</mn><mo>-</mo><mrow><mn>0.0008406944</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.9633182</mn></mrow><mo>+</mo><mrow><mn>0.03132404</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.05263075</mn></mrow><mo>-</mo><mrow><mn>0.04412857</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.1712351</mn></mrow><mo>-</mo><mrow><mn>0.1531988</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>-</mo><mn>0.01670809</mn><mo>-</mo><mrow><mn>0.009586891</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.01254876</mn></mrow><mo>-</mo><mrow><mn>0.06168799</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>-</mo><mn>0.009968993</mn><mo>+</mo><mrow><mn>0.009517347</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>-</mo><mn>0.005665266</mn><mo>-</mo><mrow><mn>0.01805476</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>-</mo><mn>0.000592167</mn><mo>+</mo><mrow><mn>0.007689914</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" 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height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.0003381264</mn></mrow><mo>+</mo><mrow><mn>0.00000327865</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>-</mo><mn>0.0001684702</mn><mo>+</mo><mrow><mn>0.00004050884</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.000009928597</mn></mrow><mo>-</mo><mrow><mn>0.00009819612</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.000002819018</mn></mrow><mo>+</mo><mrow><mn>0.0000503193</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>0.000007896499</mn></mrow><mo>-</mo><mrow><mn>0.000002379687</mn><mo>⁢</mo><mstyle><mspace width="0.6em" 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/></mstyle><mo>⁢</mo><mn>0.00003454319</mn></mrow><mo>-</mo><mrow><mn>0.00001452121</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow><mo>-</mo><mn>0.00002156384</mn><mo>-</mo><mrow><mn>0.000002776993</mn><mo>⁢</mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo>⁢</mo><mi>i</mi></mrow></mrow></mrow></mrow></math></maths>

0066These coefficients are those for an equalizer impulse response, b[n], which includes a micro-reflection at a delay that is approximately two times the symbol period, T_s. The symbol period is the inverse of the symbol rate R_s. A common DOCSIS symbol rate is R_s=5.12 Msps, therefore the symbol period T_s=195.3 ns. Since this is a T-spaced equalizer implementation, the delay associated with each tap is 195.3 ns. Therefore, the delay associated with the micro-reflection is 390.6 ns.

0067<figref idref="DRAWINGS">FIG. 19</figref> plots the magnitude of the impulse response h_eqand reveals that there is an echo or spike <b>70</b> beginning at approximately tap <b>18</b>, two symbol periods later from the main tap <b>16</b>. Based on existing techniques of estimating micro-reflection delay, one could estimate the cable length to be approximately 167 ft. of coaxial cable, but the present disclosure uses digital signal processing (DSP) in the time domain to improve the accuracy of the cable length estimate.

0068First, the channel response A(z) may be estimated by receiving the coefficients from pre-equalizer <b>60</b> to form a time domain response <b>66</b>, as illustrated in <figref idref="DRAWINGS">FIG. 18</figref>, that may be converted to the z-domain and inverted using equation 14. However, as illustrated by the z-plane plot illustration in <figref idref="DRAWINGS">FIG. 20</figref>, of the coefficients of b[n] may produce a transfer function B(z) having many zeroes falling outside the unit circle, which makes inverting h_eqproblematical. In such a circumstance, calculating a minimum phase version of b[n] using Equation 6, the MATLAB ploystab( ) function, will produce a z-domain response more suitable for inversion, but yet still retain the characteristics of the channel. <br /><i>h</i>_min=polystab(<i>h</i>_eq)*norm(<i>h</i>_eq)/norm(polystab(<i>h</i>_eq))  (6)<br /> The minimum phase (MP) filter, h_min, illustrated in <figref idref="DRAWINGS">FIG. 21</figref> now has all the zeros located within the unit circle, thus invertible.

0069A comparison of the magnitude of the frequency responses of b[n] and h_minshow that they are substantially identical, as seen in <figref idref="DRAWINGS">FIG. 22</figref>, confirming that h_minmay be used as a substitute for b[n]. Note that the coefficients of both b[n] and h_minare a time domain representation of the equalizing filter and the channel respectively. The frequency domain responses may be obtained by performing the Discrete Fourier Transform (DFT) on the time domain responses (possible using zero-padding), which is illustrated in <figref idref="DRAWINGS">FIG. 22</figref>. The frequency response is just as revealing at the time domain. The in-band ripple amplitude will be proportional the micro-reflection level. The number of complete waveforms is proportional to the delay, which in this case there are approximately two. Note to convert a frequency domain response back to the time domain, the Inverse Discrete Fourier Transform (IDFT) would be used.

0070Given the magnitude responses are equivalent, h_mincan be inverted to obtain the channel response A(z) and converted to the time domain [in tap samples n] as described above. To improve the resolution, interpolation is used to increase the samples between the taps, in this case by a factor of 32, illustrated in <figref idref="DRAWINGS">FIG. 23</figref>. It can be observed from this figure that the peak <b>80</b> associated with the micro-reflection does not occur precisely at 2, in fact the peak is approximately 6/32T less than 2T. Translating this to cable length means that the estimated cable length is 151.3 ft. instead of 167.0 ft. predicted earlier.

0071<figref idref="DRAWINGS">FIG. 24</figref> generalizes the procedure just described. Specifically, a method <b>90</b> for estimating a distance to a cause of a micro-reflection may begin by receiving in step <b>92</b> coefficients to be used by a pre-equalizer to compensate for a micro-reflection. The pre-equalizer response may preferably be calculated in a CMTS of a head end, such as the head end <b>22</b> of FIG. <b>4</b>. In step <b>94</b>, it may optionally be determined whether the coefficients produce a response that can be inverted in the z-domain. If not, in step <b>95</b> the coefficients may be modified to produce a response capable of inversion, such as by using a polystab function as previously described, and then the response of the modified coefficients may be inverted in step <b>96</b>. If, conversely, the coefficients are determined to produce a response capable of inversion in step <b>94</b>, the response can simply be inverted in step <b>96</b>. It should be understood that in some embodiments, step <b>94</b> may be omitted and the coefficients may be modified by step <b>95</b> in all cases, and the modified coefficients simply used to produce a response that is inverted in step <b>96</b>. Once the response is inverted, it is converted back to the time domain and interpolated in step <b>98</b>. From the interpolated response, a better estimate of the micro-reflection's delay, and hence the distance to the micro-reflection, is calculated in step <b>99</b>.

0072Variations on the foregoing time-domain approach to measuring micro-reflection delays, and distances to the cause of a micro-reflection are also possible. As one example, an input stimulus can be provided to a minimum phase, Finite Impulse Response (FIR) digital filter that has been designed with the equalizer's converged set of coefficients, and samples its output at a resolution necessary to achieve enhanced resolution and accuracy of the micro-reflection delay estimate. For example, an impulse δ[n], input may be passed through a filter to estimate a converged response, resulting in an output a[n], which may be examined in a similar manner to the coefficients of the method described in <figref idref="DRAWINGS">FIG. 23</figref>. It should be understood that the input need not be limited to an impulse.

0073As another example, a cross-correlation function may be applied to an impulse stimulus and a filter output whose implementation is based on the FIR filter just described. The filter input may again be an impulse stimulus, which is provided to the filter and the cross-correlation function simultaneously. The cross-correlation function compares its two inputs, providing a spike indicating the delay at which the two signals become aligned or correlated. Resolution will be adjusted as necessary to achieve enhanced accuracy of the micro-reflection delay estimate. Note again that the input need not be limited to an impulse.

0074It should be understood that the foregoing techniques may be embodied in any hardware implementation, and in particular may be implemented by the processor <b>38</b> in a CMTS of a head end <b>22</b> shown in <figref idref="DRAWINGS">FIG. 4</figref>, such that when queried, the CMTS can provide an accurate estimation of the distance to a cause of a micro-reflection by processing the same coefficients that it calculates and provides to a pre-equalizer to mitigate the micro-reflection.

0075The approaches disclosed above characterize micro-reflection delay from the pre-equalization process, but the disclosure is not limited to this approach. For example, any given network may have many terminal devices including cable modems and set top boxes that can be used to detect and isolate micro-reflection impairments by serving as network test points. Leveraging the CATV network infrastructure for diagnostics and repair is preferable to transporting and connecting test instrumentation into the network for manual measurements.

0076The terms and expressions that have been employed in the foregoing specification are used therein as terms of description and not of limitation, and there is no intention, in the use of such terms and expressions, of excluding equivalents of the features shown and described or portions thereof, it being recognized that the scope of the claimed subject matter is defined and limited only by the claims that follow.